SemanticDecisionProcedures forSomeRelevantLogics · 2019. 10. 27. · In Relevant Logics and their Rivals [12, pages 399–406], Richard Routley (as he was then known) purported to

Semantic Decision Proceduresfor Some Relevant Logics

R BP P, L T U

[email protected]

Received by Greg RestallPublished July 1, 2003

http://www.philosophy.unimelb.edu.au/ajl/2003

c© 2003 Ross Brady

Abstract: This paper proves decidability of a range of weak relevant logics using de-cision procedures based on the Routley-Meyer semantics. Logics are categorized as-logics, for those proved decidable using a filtration method, and -logics, for thoseproved decidable using a direct (unfiltered) method. Both of these methods are setout as reductio methods, in the style of Hughes and Cresswell. We also examine someextensions of the -logics where the method fails and infinite sequences of worlds canbe generated.

In Relevant Logics and their Rivals [12, pages 399–406], Richard Routley (as hewas then known) purported to have established the decidability of a range ofweak relevant logics. He first claimed to have proved decidability for the sys-tem , on pp. 401-2, using a filtration method upon the Routley–Meyer seman-tics, which yielded the finite model property for . After warning that themethod would by no means extend to all the systems with semantic postulatesgiven in Chapter 4 and 5 of [12], he extended the result to the postulate Raaa,for the axiom A & (A → B) → B, the postulate Raa∗a, for A → ∼A → ∼A,and also to the postulate, if Rabc then Rac∗b∗, for A → ∼B → .B → ∼A.There are also a couple of less significant postulates given. Routley then exam-ined the use of a second filtration, but realized its shortcomings in establishingpostulate 2 — if a ≤ b and Rbcd then Racd — for the finite model. Hebriefly examined a third filtration, realized that it would not yield a decidabil-ity result without deductive closures on the worlds of the finite models, andhence suggested using operational semantics instead of the Routley–Meyer se-mantics.However, Routley had erred in two places in his first filtration. Fortunately

for those of us interested in weak relevant logics and Routley–Meyer-style se-mantics, his second filtration can still be used and his results can be revived andextended using the simplified semantics of Priest and Sylvan [10] and Restall

“Semantic Decision Procedures for Some Relevant Logics”, Australasian Journal of Logic (1) 2003, 4–27

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http://www.philosophy.unimelb.edu.au/ajl/2003 5

[11], which has subsequently become available. The object of §3 of this paperis to show how this is to be done. In the process, I do not want to detract fromwhat has otherwise been an outstanding volume, of which all of us in the fieldmake heavy use.It must also be said that Fine had first proved the decidability of weak

relevant logics without A → B → .B → C → .A → C and A → B → .C →A → .C → B in [6], pp. 365–8, using his own semantics which embraces boththeories and prime theories. The above postulate 2 simplifies to — (ii) ift ≤ u then (t ·v) ≤ (u ·v) [6, page 348] — which enables the decidability proofto go through without anything resembling the simplified semantics of Priest,Sylvan and Restall.Moreover, a different range of systems have been shown decidable using

proof-theory in Brady [3,4,5], viz. the contraction-less logics, , and ,by using Gentzen systems based on the work of Dunn [1, pages 381–391] andGiambrone [7]. This suggests that there may be a semantic method, based ona Routley-Meyer semantics, for establishing the same result, which might thenextend to other systems. We will show this in §5 of the paper, indicating whatgoes wrong with some of these extensions.

1 T LWe present axiomatizations for the main logics referred to in this paper: , ,, , , , , , , , , , , , and , togetherwith their disjunctive rule extensions. We take as primitives: ∼, &, ∨, →, andwe consider the following axioms, rules and meta-rule, and the class of logicsconstructed using them, in Figures 1 and 2.1 (Each of the subtracted axiomsand rules are redundant in the respective system.)For any logic , let d be + 1. For each of the listed logics without

12, = d, i. e., they have the same set of theorems. (See Slaney [13] and [14]for this result.)

We distinguish two kinds of logics for the purposes of this paper : An -logicis one of the following: d (or ), d (or ), d (or ), d, d, d, d,d. These will used in §3 for filtrations and in §4 for the reductio methodbased on these.A -logic is one of the following : d (or ), d, d, d (or ), d,

d, d (or ), d, d, d (or ). These will be used in §5 forthe reductio method without filtrations. Since simplified semantics are usedthroughout, disjunctive rules are always required.

1We use ‘’ to represent the logic , without the two -axioms, 16 and 17, but with theother half of classicality, 5.


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1 A → A2 A & B → A3 A & B → B4 (A → B) & (A → C) → .A → B & C5 A → A ∨ B6 B → A ∨ B7 (A → C) & (B → C) → .A ∨ B → C8 A & (B ∨ C) → (A & B) ∨ (A & C)9 ∼∼A → A10 A → ∼B → .B → ∼A11 (A → B) & (B → C) → .A → C12 A ∨ ∼A13 A → ∼A → ∼A14 (A → .A → B) → .A → B15 A & (A → B) → B16 A → B → .B → C → .A → C17 A → B → .C → A → .C → B18 A → .A → B → B1 A,A → B ⇒ B2 A,B ⇒ A & B3 A → B, C → D ⇒ B → C → .A → D4 A → ∼B ⇒ B → ∼A5 ∼A, A ∨ B ⇒ B1 If A ⇒ B then C ∨ A ⇒ C ∨ BFigure 1: Axioms, Rules and Meta-Rule

2 T S SPriest and Sylvan [10] introduced a simplified version of the Routley–Meyersemantics for the logic + and then , the essential feature being the removalof the defined ordering relation ‘≤’ for the reduced semantics and the use ofthe following special truth-condition for evaluating entailments A → B at thebase world T , viz.

• I(A → B, T) = T iff, for all b ∈ K, if I(A,b) = T then I(B, b) = T .The usual Routley–Meyer truth-condition applies forA → B for worlds a 6= T .Then, Restall [11] extended this simplified semantics to a wide range of rel-

evant and other logics. However, he did not use the special truth-condition forA → B, but instead added the semantic postulate, RTab ⇔ a = b, which sim-plifies some of the other postulates, together with the corresponding sound-ness and completeness arguments. For our purposes, it is Restall’s form of thesemantics that works best and so we set it out for our logics of interest. Westart with the logic d, which has the same theorems as .




1–9 + 1–4 + 12 + 5 + 10 − 4 + 12 + 5 + 11 + 12 + 13 − 12 + 14 + 15 + 5 + 18 + 16 + 17 − 3 + 12 + 5 + 18 − 17

Figure 2: Systems

A model structure (ms) consists of the concepts 〈T, K, R, ∗〉, where K isa set of worlds, the base world T ∈ K, R is a 3-place relation on K, and ∗ is a1-place function on K, subject to the semantic postulates: for a, b ∈ K:

1 a∗∗ = a.

2 RTab ⇔ a = b.A -model is a ms with valuation v, which assigns a truth-value T or F to eachsentential variable for each world. Each valuation is uniquely extended to aninterpretation I by induction on formulae, as follows: for a ∈ K, sententialvariable p, and formulae A, B:

(i) I(p, a) = v(p, a).

(ii) I(∼A,a) = T iff I(A,a∗) = F.

(iii) I(A & B, a) = T iff I(A,a) = T and I(B, a) = T .

(iv) I(A ∨ B, a) = T iff I(A,a) = T or I(B, a) = T .

(v) I(A → B, a) = T iff, for all b, c ∈ K, if Rabc and I(A,b) = T thenI(B, c) = T .

A formula A is true in a -model iff I(A, T) = T , and A is valid in a ms iff A istrue in every -model on that ms. A is valid in the simplified semantics for iffA is valid in every ms.




T 1 For formulae A, if A is a theorem of d (or ) then A is valid inthe simplified semantics for .P The usual soundness theorem follows as in [12], but with 2 simpli-fying the proof of the Entailment Lemma (Lemma 4.2 on page 302) for reducedmodelling. For 1, see Relevant Logics and their Rivals [12, pages 336-7], orPriest and Sylvan [10, pages 220-1]. ]

T 2 For formulae A, if A is valid in the simplified semantics for then A is a theorem of d (or ).P The completeness theorem is generally standard [12], but with twoessential differences. First, the canonical base world TB is constructed so thateach rule of d is closed in it, with help from 1. (This construction was usedin Brady [2, pages 252–256], with quantifiers.) Second, to prove RBTBab ⇔a = b in the canonical model structure, Restall introduces (for +) [11, page490], a copy of TB, called T ′B, which has the same members as TB, but satisfiesthe above equivalence for a, b ∈ KB. RBabc is then defined normally for a 6=T ′B. Then, in proving Ic(A → B, a) = T iff A → B ∈ a, for the canonicalinterpretation Ic, we need to consider the special case, a = T ′B. However, for ,with negation, we also need a copy of T∗B, viz. T∗′B , so as to satisfy 1.2 RBT∗′B abis just defined as RBT∗Bab, T∗′B and T∗B being regarded as distinct elements ofKB. ]We extend these soundness and completeness results to the -logics and -logics, defined in §1. Figure 3 lists the semantic postulate corresponding toeach additional axiom and rule.

T 3 For formulae A, if A is a theorem of one of the -logics or -logics L then A is valid in the simplified semantics for L.P Restall [11, pages 484–6] showed soundness for 11, 14, 15, 16, 17and 18. After introducing an extra semantic primitive, ‘≤’, with postulatesand valuation condition below, Restall [11, pages 506–7], showed soundness for10, 12 and 13. Soundness for 5 follows easily. ‘≤’ is a 2-place relation on K,satisfying the following: For a, b ∈ K, and sentential variables p,

13 a ≤ b ⇒ b∗ ≤ a∗14 a ≤ b & Rbcd ⇒ Racd.3 If a ≤ b and v(p, a) = T then v(p, b) = T .

The theorem, ‘if a ≤ b and I(A,a) = T then I(A,b) = T ’, then follows byinduction on formulae A [12, page 302].

2Such an addition as T∗′B was used by Giambrone and Meyer [8, pages 11–12].3Restall breaks this up into two cases: a = T and a 6= T , but this does not appear to be

needed here.




A A R C P10 A → ∼B → .B → ∼A 3 Rabc ⇒ Rac∗b∗11 (A → B) & (B → C) → .A → C 4 Rabc ⇒ ∃x ∈ K, Rabx & Raxc12 A ∨ ∼A 5 T∗ ≤ T13 A → ∼A → ∼A 6 (i) Raa∗a, for a 6= T ,

and (ii) T∗ ≤ T14 (A → .A → B) → .A → B 7 Rabc ⇒ ∃x ∈ K, Rabx & Rxbc15 A & (A → B) → B 8 Raaa16 A → B → .B → C → .A → C 9 Rabz & Rzcd ⇒

∃x ∈ K, Racx & Rbxd17 A → B → .C → A → .C → B 10 Rabz & Rzcd ⇒

∃x ∈ K, Rbcx & Raxd18 A → .A → B → B 11 Rabc ⇒ Rbac5 ∼A, A ∨ B ⇒ B 12 T ≤ T∗

[When 12 and 5 are included, replace 6 (ii)and 12 by T = T∗ and simplify 6 to Raa∗a.]

Figure 3: Semantic Postulates Corresponding to Axioms and Rules

However, we only require this process for T∗ ≤ T and T ≤ T∗. In whichcase, 13 is automatically satisfied, given 1, and 14 reduces to the two re-spective forms, given 2:

14 T∗ ≤ T ⇒ RT∗aa14 T ≤ T∗ & RT∗ab ⇒ a = bSo, in the event that T∗ ≤ T is required, i. e. for 12 and 13, we add:

15 RT∗aa

and the valuation condition:

1 If v(p, T∗) = T then v(p, T) = T , for all p

and, in the event that T ≤ T∗ is required, i. e. for 5, we add the postulate:

16 RT∗ab ⇒ a = band the valuation condition:

2 If v(p, T) = T then v(p, T∗) = T , for all p

though these are not required on their own, for the indicated logics. ]




T 4 For formulae A, if A is valid in the simplified semantics for oneof the -logics or -logics L then A is a theorem of L.P Completeness is somewhat more complicated because of the split def-inition of the canonical relation RLabc, for the logics L. We also define a ≤ b,for a, b ∈ KL, as a ⊆ b.For the logics, d, d, d, d, d, d, d, d, d, d, d,

d, dand d, this is relatively straight-forward with postulates 3, 4,6, 7, 8, 9 and 10 requiring additional checking when a = T ′L, as in Restall[11, pages 492–494, 507–8]. For 9 and 10, we also need to check b = T ′L andz = T ′L, but b = TL and z = TL do the required work in each case. (Note that8 is required for 7, but 15 is derivable from 14 anyway.) Also, we can seethat RLT∗′L aa holds for logics with 12, since T∗L ⊆ TL and RLTLaa. Further, forlogics with 12 and 5, we put T∗′L = T ′L since T∗ = T .However, there is a special definition for RL for logics L containing 18.

We will assume, for the purposes of the logic d, that 10–15 and 5 areincluded in L, as there is much interlinkage between the various axioms andthe rule in re-establishing the postulates. Also, for the purposes of d, wewill assume that 10 is included in L. We define RL as:

RLT′Lbc ⇔ b = c, RLaT ′Lc ⇔ a = c, RLabT ′L ⇔ a = b∗

RLabc is then defined normally for a 6= T ′L, b 6= T ′L and c 6= T ′L. In checkingthe postulates, 3 and 11 are fine, 8 and the unrestricted 6 use T∗′L = T ′L, 4uses 3 and 8, 7 uses 6 and 8, and 9 and 10 both use 3 and 11.In checking the proof of ‘Ic(A → B, a) = T iff A → B ∈ a’, we show that

A → B ∈ a iff, for all b, c ∈ KL, if RLabc and A ∈ b then B ∈ c. In the casea 6= T ′L, A → B ∈ T ′L iff, for all b ∈ KL, if A ∈ b then B ∈ b, as shown in [10,11]for +. For the case b = T ′L, if A → B ∈ a, a = c, A ∈ T ′L then B ∈ c, sinceA → .A → B → B is a theorem of L and a is a TL − L-theory.4 The conversedirection is no problem since TL will always suffice for T ′L. (Also see Restall [11,page 485].) In case c = T ′L, if A → B ∈ a, a = b∗, A ∈ b then B ∈ T ′L, since∼B → .A → B → ∼A is a theorem. ]3 T F MWe followRelevant Logics and theirRivals [12, pages 399–403] in setting up Rout-ley’s first filtration and show up the two defects in the argument. We thenmove onto the second filtration, but employing the simplified semantics of §2for the systems indicated. Generally, the filtration method finitizes the set ofworlds of a model by setting as equivalent worlds ones that evaluate a certainfinite set X of formulae the same way. This set X is generally the set of subfor-mulae of a given formula under test, closed under the negation of unnegatedformulae. The idea here is to show that the logic concerned has the finite

4For the definition of a T − L-theory, see [12, page 306].




model property, i. e. that each non-theorem can be falsified in a finite model.This yields decidability for the logic by a standard argument [12, pages 401-2].For an -logic L, we define an appropriate closure X of a formula A as the

set of all subformulae of A closed under the condition: if B is a member ofX with its principal connective not being a negation then ∼B is a member ofX. We consider a Routley-Meyer model 〈T,O, K, R, ∗,≤, v〉, where K is a set ofworlds, T ∈ O, O ⊆ K, R is a 3-place relation on K, ∗ is a 1-place function on K,≤ is a 2-place relation on K, subject to the set of postulates below for the logicL, for a, b ∈ K,

1 a ≤ a,

2 a ≤ b and Rbcd ⇒ Racd,3 a ≤ b ⇒ b∗ ≤ a∗,4 a = a∗∗,

5 a ∈ O, Rabc ⇒ b ≤ c,6 (∃x ∈ O)Rxaa,

(plus any extra postulates appropriate to the -logic L) and v is a valuationassigning T or F to sentential variables p at each world a ∈ K, subject to thecondition: for sentential variables p, if a ≤ b and v(p, a) = T then v(p, b) = T .We define, for worlds a, b ∈ K and appropriate closure X, a =X b for

(∀B ∈ X)(I(B, a) = I(B, b)). We call such a and b X-equivalent. We note thata =X b iff a∗ =X b∗. We defineX-equivalence classes ǎ(X) as {b ∈ K : a =X b},generated by the element a. We call these classes X-worlds and form the setǨ of such worlds, i. e. Ǩ(X) = {ǎ(X) : a ∈ K}. We will standardly leave off‘(X)’ to simplify terminology. Further, Ǩ is finite, as each X-world represents avaluation of the set of formulae in X, and so if X has n formulae then there areat most 2n valuations of X and hence at most 2n members of Ǩ.We go on to define a (finite) X-model M̌ in Ǩ which is Routley’s first filtra-

tion: 〈Ť , Ǒ, Ǩ, Ř, ∗̌, ≤̌, v̌〉. Ť is already given, ǎ∗̌ =df (a∗)̌, and Ǒ, Ř, ≤̌, and v̌are given as follows:

• ǎ ∈ Ǒ iff (∀B ∈ X)I(B → B, a) = T ,• Řǎb̌č iff (∀B, C ∈ X)(B → C ∈ X & I(B → C, a) = T & I(B, b) = T ⇒

I(C, c) = T),

• ǎ≤̌b̌ iff (∀B ∈ X)(I(B, a) = T ⇒ I(B, b) = T),• v̌(p, ǎ) = T iff v(p, a) = T and p ∈ X.




The first problem is that the definition of ǎ ∈ Ǒ does not seem independentof the choice of the element a in ǎ. One could have a =X b, I(B → B, a) = Tand I(B → B, b) = F, for some B ∈ X such that B → B 6∈ X, since the formulaB can be evaluated at worlds outside of ǎ.The second problem is in checking 5, we again seem to need B → B ∈ X,

when B ∈ X. We let B ∈ X and I(B, b) = T , and, since ǎ ∈ Ǒ, I(B → B, a) = T .Since Řǎb̌č we would need B → B ∈ X to derive I(B, c) = T , as required forb̌≤̌č.One could try replacing the definition of ǎ ∈ Ǒ by (∀B → B ∈ X)I(B →

B, a) = T . This solves the first problem but 5 still does not follow. One couldadditionally try putting (∀B ∈ X)(B → B ∈ X & I(B, a) = T ⇒ I(B, b) = T)for ǎ≤̌b̌. This then solves the two problems but creates a new one for 2.Clearly, we cannot add the condition ‘if B ∈ X then B → B ∈ X’ without losingthe finitude of X.We could try to remove Ǒ by using reduced models, i. e. by putting Ǒ = Ť ,

but 5 is still a problem for the original definition of ≤̌ and 2 is still a prob-lem for the revised definition of ≤̌. So, we consider the simplified semantics,without Ǒ and ≤̌, but retaining the defintion of Ř. This does overcome theproblems with the initial postulates, and does provide finite modelling for suchsystems as d and d, as set out in [12, pages 400–3], but does not cope withany of the existential postulates, such as 4 and 7 of §2, which would yieldsystems such as d and d.So, we move on to the second filtration [12, pages 403-4], which is defined

as for the first filtration, but with the following definitions of Ǒ, Ř and ≤̌:

• ǎ ∈ Ǒ iff (∃a ∈ ǎ)a ∈ O,

• Rǎb̌č iff (∃a ∈ ǎ)(∃b ∈ b̌)(∃c ∈ č)Rabc,

• ǎ≤̌b̌ iff (∃a ∈ ǎ)(∃b ∈ b̌)a ≤ b.

However, as pointed out in [12, page 404], there is a problem with ensuringthat the two bs in 2: a ≤ b & Rbcd ⇒ Racd, are the same after the def-initions of ǎ ≤ b̌ and Řb̌čď are applied. We overcome this problem withthe adoption of the simplified semantics, thus dropping ‘O’ and ‘≤’ (exceptfor logics containing 12 but not 5). Moreover, there is a similar problemwith the two postulates ‘Rabz & Rzcd ⇒ (∃x ∈ K)(Rbcx & Raxd)’ and‘Rabz & Rzcd ⇒ (∃x ∈ K)(Racx & Rbxd)’ in ensuring that the two zs arethe same after the definitions of Řǎb̌ž and Řžčď have been applied. So, weproceed to apply the second filtration to the -logics, d, d, d, d, d,d, d and d, which do have simplified semantics and do not use thesetwo postulates.There is another similar problem for the postulate 2 of §2, viz. RTab ⇔

a = b, in that if ŘŤ ǎb̌ is assumed then, for some a, b ∈ K, RTab may not holdfor the specific base world T in Ť . We overcome this by distinguishing in Ǩ




between Ť and Ṫ , where Ť is {T } and Ṫ is {b ∈ K : T =X b}− {T }. We also add Ť ∗̌which is {T∗}, to be distinguished in Ǩ from Ṫ ∗̌ which is {b ∈ K : T∗ =X b}−{T∗}.[If Ṫ is null, then delete Ṫ and Ṫ ∗̌ throughout the following argument.]The base world in the second filtration is then taken to be Ť . We put Ť ∗̌∗̌

as Ť , and Ṫ ∗̌∗̌ as Ṫ . We will continue to use ǎ to represent elements of Ǩ anda ∈ ǎ to represent elements of K which are members of ǎ. For the definitionsof ŘŤ ǎb̌, ǎ≤̌b̌ and v̌(p, ǎ), we let the forms a ∈ ǎ, etc. hold when T ∈ Ť andT ∈ Ť ∗̌, and for Ṫ and Ṫ ∗̌ in accordance with their set-theoretic memberships.

L 1 Whenever b ∈ ǎ, b∗ ∈ ǎ∗̌, for any element ǎ in Ǩ.P For ǎ, which is neither Ť , Ṫ , Ť ∗̌, nor Ṫ ∗̌, a =X b and hence a∗ =X b∗.Then b∗ ∈ a∗̌ and, since ǎ∗̌ =df a∗̌, b∗ ∈ ǎ∗̌. If b ∈ Ť then b = T , b∗ = T∗ andb∗ ∈ Ť ∗̌. If b ∈ Ṫ then b =X T and b 6= T . Hence b∗ =X T∗ and b 6= T∗, andthus b∗ ∈ Ṫ ∗̌. If b ∈ Ť ∗̌ then b = T∗, b∗ = T , b∗ ∈ Ť and b∗ ∈ Ť ∗̌∗̌. If b ∈ Ṫ ∗̌then b =X T∗ and b 6= T∗, b∗ =X T and b∗ 6= T , b∗ ∈ Ṫ and b∗ ∈ Ṫ ∗̌∗̌. ]

T 5 Given a model M, 〈T, K, R, ∗,≤, v〉, of the simplified semanticsof one of the -logics L, the second filtration X-model M̌: 〈Ť , Ǩ, Ř, ∗̌, ≤̌, v̌〉, asdefined above and with appropriate semantic postulates for L, is also a modelof L. (‘≤̌’ is included only for logics with 12 but not 5.)P We check the semantic postulates, 1–8, 11 and 12, listed in §2 forthe logics L. For the most part, [12, pages 400–3], is followed.1. For ǎ, which is neither Ť , Ṫ , Ť ∗̌, nor Ṫ ∗̌, ǎ∗̌∗̌ = (a∗∗)̌ = ǎ. The cases for Ť ,Ṫ , Ť ∗̌, and Ṫ ∗̌are clear from the above definitions.2. Let ŘŤ ǎb̌. Then, by definition, RTab, for some a ∈ ǎ and b ∈ b̌, andhence a = b. Then ǎ = b̌. Conversely, let ǎ = b̌. Since RTaa, ŘŤ ǎǎ andhence ŘŤ ǎb̌.3. For ǎ, b̌ and č, which are neither Ť , Ṫ , Ť ∗̌, nor Ṫ ∗̌, let Řǎb̌č. Then Rabc, forsome a ∈ ǎ, b ∈ b̌, c ∈ č, and hence Rac∗b∗. Then Řǎ(c∗)̌(b∗)̌ and Řǎč∗̌b̌∗̌.By Lemma 1, this result extends to the four additional cases.4. Let Řǎb̌č. Then Rabc, for some a ∈ ǎ, b ∈ b̌, c ∈ č and hence, for somex ∈ K, Rabx and Raxc. Let x̌ be the member of Ǩ such that x ∈ x̌. Then Řǎb̌x̌and Řǎx̌č for this x̌.5. Since T∗ ≤ T , Ť ∗̌≤̌Ť , by definition of ‘≤̌’.6. For part (i), let ǎ 6= Ť , then any such element a of K, such that a ∈ ǎ, isnot identical with T . Hence, for this a, Raa∗a and, by Lemma 1, Řǎǎ∗̌ǎ. Part(ii) is clear from 5, when required.7. Similar to 4.8. Let ǎ ∈ Ǩ. For any a ∈ ǎ, Raaa and hence Řǎǎǎ.11. Let Řǎb̌č. Then Rabc, for some a ∈ ǎ, b ∈ b̌, c ∈ č, and hence Rbac andRb̌ǎč.




12. We just consider Ť = Ť ∗̌, in view of the logics L. Since T = T∗, {T } = {T∗}and Ť = Ť ∗̌. For logics with 12 but not 5, we check the additional postulate,15, and the valuation condition 1.15. Since RT∗aa, by definition, RT ∗̌ǎǎ.1. Let v̌(p, Ť ∗̌) = T and Ť ∗̌≤̌Ť . Then v(p, T∗) = T , p ∈ X and T∗ ≤ T .Hence v(p, T) = T and v̌(p, Ť) = T . ]Let Ǐ(A, ǎ) be the interpretation which extends the valuation v̌(p, ǎ) to all

formulae A in accordance with the semantics.

C 1 For logics with 12 but not 5, with 15 and 1, for all formu-lae A, if Ǐ(A, Ť ∗̌) = T then Ǐ(A, Ť) = T .

The following theorem shows that the second filtration interpretation Ǐ(A, ǎ)over members of X takes the same value as that of all of the interpretationsI(A,a), for a ∈ ǎ.

T 6 For the modelM and the X-model M̌ of Theorem 5, for A ∈ X,Ǐ(A, ǎ) = I(A,a), for each a ∈ ǎ.P We check this result inductively on formulae in X, for a ∈ ǎ. Theargument is mostly as in [12, page 401].p: v̌(p, ǎ) = T iff v(p, a) = T , for p ∈ X.∼: Ǐ(∼A, ǎ) = T iff Ǐ(A, ǎ∗̌) = F, iff I(A,a∗) = F, by the induction hypothesis,iff I(∼A,a) = T . We use Lemma 1 to ensure that a∗ ∈ ǎ∗̌ for the given a ∈ ǎ.&, ∨: Straightforward.→: Let Ǐ(A → B, ǎ) = T . Then, for all b̌, č ∈ Ǩ, if Řǎb̌č and Ǐ(A, b̌) = Tthen Ǐ(B, č) = T . Let Rabc and I(A,b) = T , for any a ∈ ǎ and b, c ∈ K.Then, by definition, Řǎb̌č, where b ∈ b̌, c ∈ č, and, by induction hypothesis,Ǐ(A, b̌) = T and hence Ǐ(B, č) = T and I(B, c) = T . So, I(A → B, a) = T .Let Ǐ(A → B, ǎ) = F. Then, for some b̌, č ∈ Ǩ, Řǎb̌č, Ǐ(A, b̌) = T andǏ(B, č) = F. By definition, Rabc, for some a ∈ ǎ, b ∈ b̌, c ∈ č, and, byinduction hypothesis, I(A,b) = T and I(B, c) = F. Hence, I(A → B, a) = Ffor this a ∈ ǎ. However, since A → B ∈ X, I(A → B, a) = F for all a ∈ ǎ. ]The corollary shows that the splitting of the original X-worlds Ť and Ṫ doesnot affect their interpretations of formulae in X.

C 2 For A ∈ X, Ǐ(A, Ť) = Ǐ(A, Ṫ) and Ǐ(A, Ť ∗̌) = Ǐ(A, Ṫ ∗̌), giventhat Ṫ and Ṫ ∗̌ are both non-empty and are thus proper elements of Ǩ.P By Theorem 6, Ǐ(A, Ť) = I(A, T). There is an element a ∈ Ṫ suchthat a =X T and a 6= T , and thus I(A, T) = I(A,a), and, by Theorem 6,Ǐ(A, Ṫ) = I(A,a). So, Ǐ(A, Ť) = Ǐ(A, Ṫ). Ǐ(A, Ť ∗̌) = Ǐ(A, Ṫ ∗̌) follows similarly,since, by Lemma 1, a∗ ∈ Ṫ ∗̌. ]




T 7 Each -logic d, d, d, d, d, d, dand dis decid-able.P Each of the logics have the finite model property and are hence decid-able. To show this, let A be a non-theorem of one of the -logics L. Then, byTheorems 2 and 4, A is invalid in the simplified semantics for L and I(A, T) =F, for some model M. Let X be the appropriate closure of the formula A. ByTheorems 5 and 6, there is a second filtration X-model M̌ such that Ǐ(A, Ť) =F, since A ∈ X. So, A is falsified in the model M̌, which is finite since the set Ǩof X-worlds is finite. Decidability then follows [12, pages 401–2]. ]The next theorem tightens this decidability argument further by showing thatthe theoremhood or non-theoremhood of a formula A in an -logic L can bedetermined just by its X-models.

T 8 For a given formula A, A is a theorem of an -logic L iff A is truein all the X-models of L, where X is the appropriate closure of A.P If A is a theorem of an -logic L then A must be true in any of theX-models, as they are models by Theorem 5 and soundness was shown by The-orems 1 and 3. IfA is a non-theorem of L then, by the argument in the proof ofTheorem 7, Ǐ(A, Ť) = F in an X-model M̌, where X is the appropriate closureof the formula A. ]The finite size of the X-models is determinable in terms of features of theformula A under test. The size of X is no more than twice the number ofsubformulae of A, taking into account the closure condition, and the num-ber of X-worlds is limited to 2n, where n is the number of formulae in X. Infact, starting with a formula A, we can identify each possible “X-world” byits set of interpretations (T or F) on the members of X, then choose Ť andŤ ∗̌ (and Ṫ and Ṫ ∗̌) and define all the relation Ř and function ∗̌ (and also ≤̌, ifnecessary) on them to determine all the “X-models”. These “X-models” and“X-worlds” within them are then pared down if semantic postulates or valua-tion or interpretation conditions fail, producing X-models that are appropriatefor the -logic L. Thus, all the possible X-models for L can be constructedfor the formula A, which, given Theorem 8, can then be shown to be invalid,if Ǐ(A, Ť) = F for some X-model, or valid, if Ǐ(A, Ť) = T for all X-models.Though a decision procedure, this is very tedious and so we adapt this to thereductio method, commonly used in other areas of logic such as modal logic.

4 T E R MIn the R M, we start with the formula A under test being as-signed F at Ť , and only construct what parts of the X-model(s) of the simplifiedsemantics that are needed to try to falsify it. If an X-model can be determined,consistent with the semantic postulates and valuation and interpretation con-ditions of the -logic L, A is invalid and hence a non-theorem of L. If no such




X-models can be found, and this can only be registered in the derivation ofcontradictions in each of an exhaustive set of X-model attempts to falsify A,then A is valid in all X-models and is hence a theorem of L.We follow Hughes and Cresswell’s reductio procedure for the modal logic

[9, pages 82-96] in setting out our reductio procedures for the -logics L, butwe need to include a rule for ∗-worlds and a rule for identifying worlds. Wealso need to distinguish between the two worlds b and c when Rabc holds, asthey behave differently. We also need to add some further rules to ensure thatthe various semantic postulates for the logics L apply.We set out the rules for the reductio procedure for our -logics L, as fol-

lows: (Rules 1–6 are common to all logics L, whilst rule 7 is special to any of theparticular logics L stronger than .) (For simplicity, we drop the ‘̌ ’-notation.)1. Put the formula A under test in a rectangle, labelled with world T , and putthe truth-value F under its main connective.We continue to use rectangles around members of X, generated fromA and

taking truth-values T or F (put under the main connectives), and we append toeach rectangle a world label. In fact, when we refer to worlds in the sequelwe will assume an associated rectangle containing some members of X withtruth-values.2. We apply the usual truth-table rules to the connectives &, with T underit, and ∨, with F under it, to get definite values for the respective conjunctsand disjuncts. We continue to apply these truth-table rules to & and ∨, whereapplicable, in any of the subsequent worlds.3. For each un-starred world a, we introduce a ∗-world a∗ so that if a formula∼B takes T (F) in world a then we put B with value F (T ) in a∗, and if a formula Btakes T (F) in world a and ∼B ∈ X then we put ∼B with F (T ) in a∗. Whether a∗is a new world or not will depend on rule 5 and, in the case of T and T∗, rule 7or 7. (Note however that, for all logics L, a∗∗ = a, and that we constructeda∗ so that all the above assignments still apply when applied from a∗ into a, aswell as applying from a into a∗.)4. In a world a, if there is an ‘→’-formula taking F then there must be worldsb and c, where we put the antecedent of the ‘→’, with a T under it, into therectangle for b, and we put the consequent of the ‘→’, with an F under it, intothe rectangle for c. We pick a point on the outside of the rectangle for a, drawarrows from this point to the rectangles for b and for c, and draw a line (slash)through the stem of the arrow towards b, indicating that b is to be treateddifferently from c in relation to both this and the next clause. We call this anR-relation between a, b and c, or just Rabc. We use a different starting pointto draw such arrows for each ‘→’-formula taking F in a. Whether either orboth of b and c are new worlds will depend on rule 5. (The rule does not haveto be applied if the requisite worlds b and c already exist with the respectivevaluations.)4. In a world a, if there is an ‘→’-formula taking T , then for each existing




R-relationship between a and worlds b and c, shown by arrows from a to bwith a slash and from a to c without a slash, both from the same point on a, ifthe antecedent of the ‘→’-formula has a T under it in the rectangle for b, thenwe put the consequent of the ‘→’, with a T under it, into the rectangle for c.4. Since the postulate RTab ⇔ a = b is common to all logics L, we in-troduce the single world a (with the two arrows from the T -rectangle to thea-rectangle) when evaluating an ‘→’-formula, with an F under it, in world T ,putting both the antecedent and consequent of the ‘→’-formula into a, withappropriate values. Whether a is a new world will depend on rule 5. (The ruledoes not have to be applied if the requisite world a already exists with the re-spective valuations.) Further, for every world b, including T , RTbb holds andso, when evaluating an ‘→’-formula, with T under it, we draw in the arrows sothat whenever the antecedent of the ‘→’-formula takes T in b the consequenttakes T in b.5. Any two worlds with the same value assignments for all the members ofX must be identified. This means that the same rectangles are used for boththese worlds and thus the same ∗- and R-relationships hold for these worlds.However, there are exceptions for T and T∗. All the worlds having the samevalues over X as for T are put identical to Ṫ , and all the worlds having thesame values over X as for T∗ are put identical to Ṫ∗. Moreover, one can get tosituations where there are no further distinct X-worlds and in the applicationof rule 4 or even rule 3 one is left to consider all the possible identificationsbetween the worlds needed for these rules and the existing worlds.5. For the purpose of trying to establish a consistent X-model, we may chooseto make an identification between worlds so that there is a consistent assign-ment of values to the formulae in the identified world. If this indeed yieldsan X-model, this will result in a restricted form of the X-model one is workingwith. We will call this a restricted X-model. If a contradiction in another worldis derived from this identification then this contradiction only applies to thisparticular identification and other identifications would have to be tried to seewhether X-models or contradictions result.6. In applying the usual truth-table rules to the connectives &, with F un-der it, and ∨, with T under it, we get three different value combinations forthe respective conjuncts and disjuncts. These yield what we call alternativeX-models. Whilst any of these alternatives would suffice as an X-model, con-tradictions would need to be established in all three to produce a general con-tradiction for the procedure at this point.

Rules 1–6 apply for the logic . For stronger logics L, we now add specialrules applicable for the particular logic L we are dealing with. We examine thevarious semantic postulates of §2 in turn. Note that different starting pointsare used for each pair of arrows representing an R-relation, drawn from anysingle rectangle.




7. For 3. Rabc ⇒ Rac∗b∗, we add the R-relation Rac∗b∗ to each existingR-relation Rabc, inserting formulae and values according to rule 4. We donot need to re-apply this rule to Rac∗b∗ as Rab∗∗c∗∗ already holds.7. For 4. Rabc ⇒ x ∈ K, Rabx and Raxc, we add a world x and the R-relations Rabx and Raxc to each existing R-relation Rabc. We insert formulaeand values according to rule 4 for both Rabx and Raxc. (The rule does nothave to be applied if the requisite world x already exists with the respectiveR-relations.)Realising that this process could run on until all X-worlds are exhausted, I

would suggest identifying x with b, if possible, in accordance with rule 5. Forthen 7 need not be re-applied to Rabb.7. For 5. T∗ ≤ T , we introduce the special ≤-relationship, indicated by asingle arrow from T∗ to T , in which every formula taking T at world T∗ is giventhe value T in world T and every formula taking F at T∗ is given the value F at T .7. For 6. Raa∗a, we add this R-relation for each world a other than T andapply rule 4. T∗ ≤ T is then added as for rule 7. If T = T∗ applies, thenidentify these worlds as well. By rule 5, this will imply the identification of Ṫand Ṫ∗, as formulae would have the same values in Ṫ and Ṫ∗.7. For 7. Rabc ⇒ (∃x ∈ K)Rabx & Rxbc, we proceed in a similar manner torule 7, but I suggest identifying x with a, for then 7 need not be re-appliedto Raba.7. For 8, Raaa, we add this R-relation for each world a and apply rule 4.7. For 11. Rabc ⇒ Rbac, we proceed in a similar manner to rule 7, and wedo not need to re-apply this rule to Rbac.7. For 12. T = T∗, we identify these worlds. As for rule 7, Ṫ and Ṫ∗ are alsoidentified. (We note that T ≤ T∗ does not arise for the logics L.)

The following theorem states the relationship between the reductio methodand theoremhood for the -logics L.

T 9 A formula A is a theorem of an -logic L if a contradiction isobtained in each of an exhaustive set of derived X-models for L that assignthe value F to A at world Ť . A is a non-theorem of the logic L if a consistentX-model for L is obtained which assigns the value F to A at world Ť .P Clearly, if a contradiction is obtained in each of an exhaustive set ofderived X-models for L that assign the value F to A at world Ť then there areno consistent such X-models for A and Ǐ(A, Ť) = T for all X-models of L.Then, by Theorem 8, A is a theorem of L. In the process of exhausting a setof derived X-models for L one must include all alternative X-models under rule6 and all restricted X-models under rules 5 and 5. On the other hand, if aconsistent X-model for L is obtained which assigns the value F to A at worldŤ then Ǐ(A, Ť) = F for this X-model and, by Theorem 8, A is a non-theoremof L. ]




5 T R M C- LWe proceed to develop a similar reductio method for the contraction-less log-ics d, d, d and d, and, except for d, systems obtained by the ad-dition of 12. A ∨ A (add ‘’ to name, prior to ‘d’) and systems obtained bythe addition of both 12 and 5. ∼A,A ∨ B ⇒ B (add ‘’ to name, prior to ‘d’).These are called the -logics in §2. As indicated in §3, the first and second fil-trations will not work for the (new) axioms 16 and 17 of , and so, we needanother approach. (Recall that = d, = d, = d and = d.We will drop the ‘d’ when naming these systems for simplicity.)We will not use a filtration method and tackle the semantics directly. Since

and have already been treated in §4, we start by focussing on . If weconsider the two semantic postulates for 16 and 17 of , viz.

9 Rabz & Rzcd ⇒ ∃x ∈ K, Racx & Rbxd, and10 Rabz & Rzcd ⇒ ∃x ∈ K, Rbcx & Raxdthere seems to be a limit to the number of re-applications of these postulates ifthe introduced worlds x are new on each occasion and there are no postulates,further than those for , to concern us, at least for the moment. In orderto apply these postulates there must be two R-relations of the form Rabz andRzcd, and the introduced x cannot play the role of the z as they stand, whichmeans that we have to look around at our original set of R-relations for otherworlds for z to equal. This means that z must ultimately come from the finiteset of worlds and R-relations that we would already have from the reductiomethod to this point and, if we insist that the introduced worlds are all new,these two postulates will not have the propensity to generate an infinite num-ber of worlds. We set this argument out more fully below.We will continue to use the simplified semantics as it will give a simpler and

more straight-forward decision procedure. Assuming the introduction of suchnew worlds for the logic , we set out the rules for the reductio procedure,much as in §4:

1. Put the formula A under test in a rectangle, labelled with world T , and putthe truth-value F under its main connective.2. We apply the usual truth-table rules, in this and subsequent worlds, to theconnectives &, with T under it, and ∨, with F under it.3. For each un-starred world a, we introduce a ∗-world a∗ so that if a formula∼B takes T (F) in world a then we put B with value F (T ) in a∗, and if a formulaB takes T (F) in world a then we put ∼B with F (T ) in a∗. a∗ is to be a newworld, unless there is already a world with all these properties. (Note as beforethat a∗∗ = a.)4. In a world a, if there is an ‘→’-formula taking F then there must be worldsb and c, where we put the antecedent of the ‘→’-formula, with a T under it,




into the rectangle for b, and we put the consequent of the ‘→’, with an F underit, into the rectangle for c. As before, we draw arrows from a point on a to therectangles for b and for c, with a slash through the arrow towards b. We callthis an R-relation between a, b and c, or just Rabc. b and c are both taken tobe new worlds, except as for rule 4 below. These worlds are new with respectto existing worlds and their ∗-worlds. (The rule is not applied if the requisiteworlds b and c already exist with the respective valuations.)4. In a world a, if there is an ‘→’-formula taking T then, for each existingR-relationship between a and worlds b and c, shown by arrows from a to bwith a slash and from a to c without a slash, both from the same point on a, ifthe antecedent of the ‘→’-formula has a T under it in the rectangle for b, thenwe put the consequent of the ‘→’, with a T under it, into the rectangle for c.4. Since, for the simplified semantics, RTab ⇔ a = b, we introduce thesingle world a (with the two arrows from the T -rectangle to the a-rectangle)when evaluating an ‘→’-formula, with an F under it, in world T , putting boththe antecedent and consequent of the ‘→’-formula into a, with appropriatevalues. Similarly to 4, the world a is new, unless such a world already exists.Further, for every world b, including T , RTbb holds and so, when evaluatingan ‘→’-formula, with T under it, we draw in the arrows so that whenever theantecedent of the ‘→’-formula takes T in b the consequent takes T in b.6. In applying the usual truth-table rules to the connectives &, with F underit, and ∨, with T under it, we get three different value combinations for therespective conjuncts and disjuncts. These yield what we call alternative models.7. We add the R-relation Rac∗b∗ to each existing R-relation Rabc, insertingformulae and values according to rule 4.7. For each existing pair of R-relations Rabz and Rzcd, we add the R-relationsRacx and Rbxd, and Rbcy and Rayd, where both worlds x and y are new(i. e. new to existing worlds and their ∗-worlds), inserting formulae and valuesaccording to rule 4. This rule is not applied when such x or y already existwith the requisite R-relations.

For the purposes of comparing this reductio procedure with one based pre-cisely on the simplified semantics for , we consider, in the proof below, theaddition of the following rule:

5. We can identify worlds a∗, introduced in rule 3, with any existing world, aswe can for the worlds b and c, introduced in rule 4, the world a, introducedin rule 4, and also the worlds x and y, introduced in rule 7, whether such anexisting world initially had the requisite properties or not.

We now state the relationship between the above reductio method and the-oremhood for , the proof of which involves a general argument concerningthe use of new worlds.




T 10 A formula A is a theorem of if a contradiction is obtainedin each of an exhaustive set of derived models that assign the value F to A atworld T . A is a non-theorem of if a consistent model is obtained whichassigns the value F to A at world T .P If a consistent model is obtained which assigns the value F to A atworld T then I(A, T) = F for this model and A is invalid in the simplifiedsemantics for . By Theorem 3 of §2, A is a non-theorem of . (Note that = d.)Let a contradiction be obtained in each of an exhaustive set of derived

models that assign the value F to A at world T . These contradictions wouldexhaust a set of alternative models, but would be based on the above rules 3,4, 4 and 7, where new worlds were chosen for a∗, b, c, a, x and y. If wewere to add rule 5 and allow identifications of these worlds with pre-existingworlds then the argument to the contradiction(s) would still apply as therewould be no changes to the existing assignments of values in the now identifiedworlds. Indeed, the identifications may induce further contradictions. Theremay also be further applications of rule 7 due to the identifications, but this isnot going to remove any derived contradictions. Also, any contradiction basedon the lack of application of the rules 3, 4, 4 and 7, due to the presenceof pre-existing worlds with the requisite properties, would still apply if suchrules were nevertheless applied to introduce new worlds or other worlds inconjunction with rule 5. Thus, we have established the contradiction(s) withinthe simplified semantics for , and hence there can be no consistent modelsfor A such that I(A, T) = F. So, I(A, T) = T for all models and A is valid inthe simplified semantics for . Hence, by Theorem 4 of §2, A is a theoremof . ]We now proceed with the decidability argument based on the reductio

method for .

T 11 is decidable, using the above reductio method.P The reductio method for builds up a set of worlds, starting withT, existentially introducing new worlds using the rules 3, 4, 4 and 7, thelatter three with new R-relations. We also introduce new R-relations using therules 4, 4 and 7. We examine the various types of world and R-relation in-troduction. Rules 3 and most of 4 are formula-based, in that the presence of aformula induces the rule, whilst rule 7 and the second part of 4 (4–2) involv-ing the R-relation RTbb are not, in that the rule applies to any existing world orwhen an R-relation or certain configuration of R-relations occurs. Rules 3 and7 are idempotent, i. e. the original world or R-relation is reached after two ap-plications of the respective rule. The rule 4–2 applies only to the pre-existingworlds b.So, the rules 3, 4– and 7 are either formula-based or idempotent or

rely on pre-existing worlds. (We also use this terminology for the associated




R-relations.) A reductio system, all of whose rules are of one of these threetypes, would be decidable as only finitely many worlds could be generated, thenumber of which would be dependent upon the number of subformulae in theformula under test, and so the number of generated R-relations would also befinite.The only rule that is likely to cause trouble as far as decidability is con-

cerned is rule 7, as it may be capable of generating, through successive appli-cations, an infinite sequence of worlds. We examine the possibilities for this.First consider the interaction of rule 4-2, which yields RTaa. Let this

supply one of the two R-relations, Rabz and Rzcd. If RTaa and Racd thenRTcx and Raxd, for some x, and Racy and RTyd, for some y. We do not applyrule 7 here as can be seen by putting x = c and y = d. Let us consider RabTand RTcc. The only way RabT can be introduced is by rule 4-2, in which case,a = b = T and this becomes an instance of the first case. Clearly, RabT is notformula-based. The problem with introducing RabT using rule 7 with ‘T ’ inthe d-position is that ultimately one must start with RTTT and RTTT , whichis also an instance of the first case above. If we try to derive RaT∗b∗ usingrule 7, one must start with something of the same form, which is again notformula-based. As a result we can assume that both the Rabz and the Rzcd aregenerated by rule 4 and/or rule 7. Rule 7 does not have a significant impactdue to its idempotence. Consider an application of rule 7 in the forms, Rabzand Rzcd ⇒ Racx and Rbxd, for some world x; and Rabz and Rzcd ⇒ Rbcyand Rayd, for some world y. Since x and y are new, there is no immediategeneration of further worlds by application of rule 7. Also, the R-relationsthat start such applications are all formula-based and so are finite in number.Since each world introduced by rule 4 is new, we would have finite strings ofthese R-relations all linked together like Rabz and Rzcd are, or like Rabz andRbcd, depending on the shape of the formulae involved. Other than the linkz, the other worlds are all different, and when rule 7 is applied this link isreplaced by other linking worlds x and y.We consider a string of linked formula-based R-relations of length n, linked

so that rule 7 can apply between successive R-relations. We assume that afinite number of worlds are generated by an exhaustive set of applications ofrule 7 to this string. We then add another linked formula-based R-relation,Rdef, to the end of the string, making it a string of length n+1, and show thatthere are only finitely many extra worlds introduced by exhaustive applicationsof rule 7.Consider R-relations of the type Rayd and Rbxd, obtained after any (finite)

number of applications of 7 to the initial string of length n, where the ‘x’ and‘y’ are worlds introduced by the last such application of 7 and ‘a’ and ‘b’ caneither be worlds from a formula-based R-relation Rabz or introduced by 7 likethe ‘x’ or the ‘y’ worlds but earlier. The R-relations Rayd and Rbxd are theonly types that can interact with Rdef using rule 7. So, in applying 7 here,we obtain Raex ′ and Ryx ′f, Ryey ′ and Ray ′f, Rbex ′′ and Rxx ′′f, and Rxey ′′




and Rby ′′f, where x ′, y ′, x ′′ and y ′′ are all new. To take this further, each ofthe worlds a, b, x and y in the first positions may link up with an R-relationwith these worlds in third position, such an R-relation being obtained from anumber of applications of 7 to the initial string of length n. Such worlds a,b, x and y would have been introduced by an application of 7, involving forexample something of form Rx̀cx for the ‘x’, where ‘x̀’ and ‘c’ are worlds from aformula-based R-relation or introduced earlier (than ‘x’) by 7. Applying 7 toRx̀cx and Rxey ′′ yields Rx̀ex ′′′ and Rcx ′′′y ′′, and Rcey ′′′ and Rx̀y ′′′y ′′, for newx ′′′ and y ′′′. There may be other subsequent occurrences of such forms Rx̀cxobtained by subsequent applications of 7, but the same structure applies. Onecan again take this further by linking up with the worlds x̀ and c. What isapparent however is that on each such linkage the worlds are introduced earlierand earlier until, by successive application, there is no such linkage at all dueto the worlds being part of a formula-based R-relation. So the whole processis finite, introducing only finitely many new worlds and R-relations. Note thatthere is no generation of further worlds by linking any of the newly introducedworlds x ′, y ′, x ′′, y ′′, x ′′′, y ′′′, etc. as they cannot appear in first positionwithout an R-relation of form Rfgh. Also note that application of 7 wouldjust give a ∗-version of this whole process.So, by an induction argument, only finitely many worlds can be introduced

by rule 7 from any string of linked formula-based R-relations of any finitelength. This addresses our only remaining concern and thus is decidable,using the reductio method. ]

C 3 and are decidable.P The reductio method for is obtained by removing rule 7 fromthat for and the reductio method for is obtained by removing rule 7from that for . Hence and are both decidable by these methods as,with the absence of 7, there is no impediment to the finiteness of the set ofintroduced worlds, as mentioned in the above proof. ]We now extend the decidability argument for to and then to the re-maining -logics.

T 12 is decidable.P We first simplify the formulae that we need to consider so that theproof can go through. As shown by Slaney [13], it suffices to consider justentailments of form A → B when determining theorems of , due to theprimeness of theorems, viz. ` A ∨ B iff ` A or ` B, and the break up ofnegated entailments, viz. ` ∼(A → B) iff ` A and ` ∼B. So, for the reductiomethod, we do not have to consider negated entailments being true at world Tnor entailments false at T∗, and so the formula-based R-relation RT∗ab, and itsequivalents RaT∗b and Rab∗T (see rule 7 below), would never arise. However,RT∗TT∗ can be derived, as can be seen from rule 7 below.




With the addition of the semantic postulate 11. Rabc ⇒ Rbac, we addthe following rule to those in the reductio method for :7. We add the R-relation Rbac to each existing R-relation Rabc, insertingformulae and values according to rule 4.As for rule 7, this rule is idempotent. However, unlike 7 which is fairlyharmless, rule 7 leads to some complications which will need to be examined.In conjunction with 7, 7 will allow the introduced worlds x and y to bepositioned in such a way as to allow 7 to be re-applied with x and y as links.Also, given 4, rule 7 will yield RaTa, which with 7 will generate an infinitenumber of worlds, as we will shortly see for d. However, in this case, 7 willprevent that from happening. Further, using 7 and 7, we get Raa∗T∗, whichalso needs examination.What we will show here is that there are no more new worlds introduced

by the addition of rule 7. New worlds can only be generated in conjunctionwith 7 where, given Rabz and Rzcd, we add Racx and Rbxd, for some worldx, and Rbcy and Rayd, for some world y. We assume that 7 is applied before7, wherever possible, as a matter of procedure. We address the three above-mentioned concerns in turn.(i) We expand the conclusions of 7 by using 7, as follows: Racx and Rxbd,Rcax and Rxbd, Rbcy and Ryad, Rcby and Ryad. To each of these forms, wecan re-apply 7, as follows: Rabz and Rczd, Rcby and Rayd, Rcby and Rayd,Rabz and Rczd, Rbaz and Rczd, Rcax and Rbxd, Rcax and Rbxd, Rbaz andRczd. We fill in the variable places in accordance with existing R-relations,taking into account applications of 7 in the process. So, any re-application of7, as a result of 7, does not yield any new worlds. Also note that applicationof 7 would just give equivalent ∗-versions of this process.(ii) We consider the use of RaTa with 7. Similarly to the case where RTaawas in the Rabz position, from RaTa and Rabc we get Racx and RTxd, andRTcy and Rayd, where by putting x = d and y = c we obtain pre-existingR-relations. Also, when we consider Rabc and RcTc, we get RaTx and Rbxc,and bTy and Rayc. Here, we put x = a and y = b to obtain pre-existingR-relations, with help from 7.(iii) In considering Raa∗T∗ with 7, from Rabc and Rcc∗T∗ we get Rac∗x andRbxT∗, and Rbc∗y and RayT∗, where by putting x = b∗ and y = a∗ we ob-tain pre-existing R-relations. However, with Raa∗T∗ in the Rabz position,we would need an R-relation of the form RT∗cd, which is not formula-based,as seen above. Unless c = T and d = T∗, RT∗cd is not generated by using7, since ultimately one of the forms RT∗cd, RcT∗d or RcdT would need to beformula-based. If we apply 7 to Raa∗T∗ and RT∗TT∗ we get RaTx and Ra∗xT∗,and Ra∗Ty and RayT∗, where we can put x = a and y = a∗.

There are no other ways that new worlds might be introduced by 7 that would




not be introduced otherwise, and so is decidable, given the argument for. ]

T 13 All the -logics are decidable.P We just need to examine the addition of 12. A ∨ ∼A and of 12together with 5. ∼A,A ∨ B ⇒ B, to , and . For 12, we add thesemantic postulate 5, together with the ordering relation ‘≤’, the valuationcondition 1 and 15. RT∗aa. We add the following rule 7 to the reductiomethod.7. We introduce the special ≤-relationship, indicated by a single arrow fromT∗ to T , in which every formula taking T at world T∗ is put with value T inworld T and every formula taking F at T∗ is put with value F at T . We also addthe R-relation RT∗aa, for all existing worlds a.For d and d, 7 does not change the finitude of the worlds. For

d, we need to examine the effect of RT∗aa on 7. Firstly, RT∗aa and Rabcyield RT∗bx and Raxc, and Raby and RT∗yc, where we can put x = b andy = c. Secondly, consider RabT∗ and RT∗cc, whereupon RabT∗ (and RaTb∗) isnot formula-based and cannot be obtained through applications of 7, unlessit is one of RTT∗T∗ or RT∗T∗T∗. The first of these is covered by RTaa and thesecond by RT∗aa above.For 5, we add 12. T ≤ T∗, which becomes T = T∗ in the presence of 5.

We modify rule 3 and add rule 7 below.3. a∗ is a new world for a 6= T .7. We put T∗ identical with T .For d and d and d, there is little change from their respective sys-tems, d, d and d. ]We next examine some of the major extensions of the -logics to see whatgoes wrong with the above decidability arguments and why they might thus beundecidable, though this argument itself will not constitute a proof as it dealswith only one method, viz. this particular reductio method. The logics , and have already been proved to be undecidable by Urquhart [15]. In fact, heproved undecidability for all logics between +15 and [15, pages 1069–70].Perhaps his methods can be extended to other systems where undecidability issuggested in this paper but not already shown.We start with d, which is d+6, where 6 is the rule,A ⇒ A → B →

B. Using simplified semantics, its corresponding semantic postulate is 17:RaTa, which in the absence of 7 does lead to the following infinite sequenceof worlds. Let Rabc. Then, since RcTc, by applying 7, we get RaTx andRbxc, for some new x. Then consider Rbxc and RcTc, in which case, RbTx ′and Rxx ′c, for some x ′. Further, Rxx ′c and RcTc yields RxTx ′′ and Rx ′x ′′c, forsome x ′′. Thus, an infinite sequence of worlds can be generated.




We next consider d and d, obtained from d by the successiveaddition of 12 and 5. With the addition of 12, we add the rule 7, whichincludes the R-relation RT∗bb, for any b. However, we can combine this withRaa∗T∗ using 7 to get Rabx and Ra∗xb, for some x, and Ra∗by and Rayb, forsome y. Using these new worlds x and y, Rxx∗T∗ and RT∗yy, and hence Rxyx ′and Rx∗x ′y, and Rx∗yy ′ and Rxy ′y, for new x ′ and y ′. We can thus generatean infinite sequence of worlds employing the rules 4, 7, 7 and 7 as theystand. For d, we similarly use Raa∗T and RTbb.We next consider any extension of d, d or d consisting of one or

more of the following axioms: A11, A13, A14, A15, with corresponding semanticpostulates: 4, 6, 7, 8. (However, for any such system L without 6 andeither without 12 or with 15, Ld = L.) These systems include , , and, where = + 11, = + 13 + 14, = + 13 + 14, =+14. 4 and 7 clearly lead to an infinite sequence of worlds by successivere-application. 6 and 8 are both of the form Raba, for some b. This formgenerally leads to an infinite progression by applying 7 as follows. Raba andRacd yields Racx and Rbxd, and Rbcy and Rayd. In turn, Raba and Racxyields Racx ′ and Rbx ′x, and Raba and Racx ′ yields Racx ′′ and Rbx ′′x ′. Theavoidance of the form Raba is one of the reasons why newness is important inrule 4.

R

[1] A. R. Anderson and N. D. Belnap, Jr., Entailment, The Logic of Relevance andNecessity, Vol. 1, Princeton U. P., 1975.

[2] R. T. Brady, “A Content Semantics for Quantified Relevant Logics II,”Studia Logica, Vol. 48 (1989), pages 243–257.

[3] R. T. Brady, “The Gentzenization and Decidability of ,” Journal ofPhilosophical Logic, Vol. 19 (1990), pages 35–73.

[4] R. T. Brady, “Gentzenization and Decidability of Some Contraction-lessRelevant Logics,” Journal of Philosophical Logic, Vol. 20 (1991),pages 97–117.

[5] R. T. Brady, “Simplified Gentzenizations for Contraction-less Logics,”Logique et Analyse, special issue, Vol. 137–8 (1992), pages 45–67.

[6] K. Fine, “Models for Entailment,” Journal of Philosophical Logic, Vol. 3(1974), pages 347–372.

[7] S. Giambrone, “+ and + are Decidable,” Journal of Philosophical Logic,Vol. 14 (1985), pages 235–254.




[8] S. Giambone and R. K. Meyer, “Completeness and ConservativeExtension Results for some Boolean Relevant Logics,” Studia Logica,Vol. 48 (1989), pages 1–14.

[9] G. E. Hughes and M. J. Cresswell, An Introduction to Modal Logic,Methuen, 1968.

[10] G. Priest and R. Sylvan, “Simplified Semantics for Basic RelevantLogics,” Journal of Philosophical Logic, Vol. 21 (1992), pages 217–232.

[11] G. Restall, “Simplified Semantics for Relevant Logics (and Some of TheirRivals),” Journal of Philosophical Logic, Vol. 22 (1993), pages 481–511.

[12] R. Routley, R. K. Meyer, V. Plumwood and R. T. Brady, Relevant Logicsand their Rivals, Vol. 1, Ridgeview, California, 1982.

[13] J. K. Slaney, “A Metacompleteness Theorem for Contraction-FreeRelevant Logics,” Studia Logica, Vol. 43 (1984), pages 159–168.

[14] J. K. Slaney, “Reduced Models for Relevant Logics Without ,” NotreDame Journal of Formal Logic, Vol. 28 (1987), pages 395–407.

[15] A. Urquhart, “The Undecidability of Entailment and RelevantImplication,” Journal of Symbolic Logic, Vol. 49 (1984), pages 1059–1073.



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SemanticDecisionProcedures forSomeRelevantLogics · 2019. 10. 27. · In Relevant Logics and their Rivals [12, pages 399–406], Richard Routley (as he was then known) purported to